Tomographic images are created from line or plane integral measurements of an unknown object at a variety of orientations. These integral measurements, which may represent measurements of density, reflectivity, etc., are then processed to yield an image that represents the unknown object. Projection data generated in this manner is collected into a sinogram, and the sinogram is processed and backprojected to create the image. Tomographic reconstruction is the technique underlying nearly all of the key diagnostic imaging modalities, including X-ray Computed Tomography (CT), Positron Emission Tomography (PET), Single Photon Emission Count Tomography (SPECT), certain acquisition methods for Magnetic Resonance Imaging (MRI), and newly emerging techniques such as electrical impedance tomography (EIT) and optical tomography.
The 3D Radon transform is the mathematical model that describes some of the true volumetric (i.e., 3D) imaging technologies. A single 3D Radon projection is a one dimensional function corresponding to a collection of integrals of the object function on planes at a given orientation and at different positions. The 3D Radon transform of the object is the collection of its 3D Radon projections at all orientations. In MRI, projection mode imaging collects samples of the 3D Radon transform. Projection mode imaging is considered a good candidate for imaging in the presence of patient motion. The 3D radon transform also underlies true 3D Synthetic Aperture Radar (SAR) imaging.
One of the most important applications of the 3D Radon transform (and of reprojection) is in the context of cone-beam tomography, which is an emerging imaging modality in medical imaging. Through specialized preprocessing methods, such as those described in U.S. Pat. Nos. 5,124,914, 5,257,183, 5,444,792 and 5,446,776, which are incorporated by reference in their entirety, cone-beam tomographic data can be mapped into samples of the 3D Radon transform. The problem of recovering the volume being imaged from these samples can then be solved in terms of 3D Radon transforms.
Regardless of how the data is collected, either by MRI, SAR, etc., the reconstruction problem for the 3D Radon transform often involves artifact correction. This artifact correction process involves the reprojection operation, which simulates the physical tomographic acquisition system. Furthermore, the complexity requirements are high for volumetric imaging, necessitating the use of fast algorithms. Indeed, the problem is so computationally demanding that true cone-beam tomography is not yet practical, although the hardware is becoming readily available.
Another important application of reprojection is in the realm of partial data problems. In many modalities, only partial data is available. Examples include 3D SAR and cone-beam tomography with any incomplete source orbit (such as single circle). The reconstruction problem in these cases requires the use of iterative reconstruction algorithms that require repeated reprojection operations, which also need significant acceleration to become practical. Reprojection can be used in conjunction with the hierarchical 3D Radon backprojection, as described in U.S. patent application Ser. No. 539,074, filed Mar. 30, 2000, in iterative reconstruction algorithms which require both reprojection and backprojection at each step.
Several methods of reprojection have been developed. Direct reprojection represents one method for reprojection in which the physical system is simulated using the defining equations for the 3D radon transform. This is a relatively slow method, which requires O (N.sup.5) computations for calculating O (N.sup.2) projections from an N.times.N.times.N sized volume.
The Separation method takes advantage of the fact that for certain distributions of the projection orientations, the 3D problem can be decoupled into a set of 2D problems. These 2D problems can then be solved using direct methods. In this way, a fast algorithm is constructed, that computes O (N.sup.2)projections from an N.times.N.times.N sized volume in O (N.sup.4) computations.
The Fourier Slice approach described in Swedish Patent Application No. 9700072-3, filed Jan. 14, 1997, works by the Fourier Slice Theorem, which states that the Fourier transform of the projections are "slices" of the Fourier transform of the object, and can thus be computed efficiently by interpolation from the 3D Fourier transform of the volume.
In the Linogram approach, a series of interpolations and resampling operations are used to transform the data into samples of the object's Fourier transform on a special grid. The object can then be recovered efficiently using Fast Fourier Transforms. This is also a fast algorithm which requires O (N.sup.3 log.sub.2 N) computations for computing O (N.sup.2) projections from an N.sup.3 volume.
In all, there is a need for fast, efficient and accurate reprojection processes for 3D images.